I followed the steps described in chapter 1 and 2 to learn about test results. Once I push the "screwed" game of live on the server, git pulls the changes and then turns the ball from blue to red due to the failing tests, just as the book says.

But every other minute it adds one more build, which obviously fails still, as the fix is not pushed to github so far. In fact, meanwhile I have lots of red balls meanwhile due to that.

Is this intended behaviour of your tutorial / Jenkins? The books should tell something about that, since that lots of (useless since unchanged) broken builds scares beginnes (at least: me).